Area of a rectangle

Question

f(x) = x² - a and g(x) = a - x², where a is a constant. Points S and T lie on the graph of g(x), points V and U lie on the graph of f(x), and the length of VU is 2a. Which of the following represents the area of rectangle STUV?

 

A. |4a³ - 4a²|

B. |2a³ - 2a²|

C. |a³ - a²|

D. |4a³|

E. |2a³|

Answer 1

If you plot f(x) you will find that it is a parabola with vertex at y=-a opening upward.

 

If you plot g(x) you will find that it is a parabola with vertex at y=a opening downward.

 

You will also find that g(x) is the reflection of f(x) across the x axis.

 

You are told that UV=2a, but there are many points U and V for which this is true.  Therefore, if the problem is to make sense, the area for all the possible rectangles must be the same.  If so, you can assume that UV is a horizontal line to make things simpler.  The rectangle sides are then horizontal and vertical.

 

Since UV=2a, that is the length of the rectangle's horizontal sides.

 

To get the length of the rectangle's vertical sides, note that when x=a, f(x)=a-a², which, due to the reflection has the magnitude of one half the length of the rectangle's vertical side.

 

So, A=length * width=|2(a-a²)||2a|=|4a³-4a²|

 

Choice (A) is correct.